3 Answers
3

Smith et al. (2009) provide a survey of the morphology of the cecal appendix. One current hypothesis is that the appendix provides "safe harbor" for symbiotic gut bacteria. Among mammals, there is a vast array of cecal appendices:

In summary:

A comparative anatomical approach reveals three apparent morphotypes
of the cecal appendix, as well as appendix-like structures in some
species that lack a true cecal appendix. Cladistic analyses indicate
that the appendix has evolved independently at least twice (at least
once in diprotodont marsupials and at least once in Euarchontoglires),
shows a highly significant (P < 0.0001) phylogenetic signal in its
distribution, and has been maintained in mammalian evolution for 80
million years or longer.

Aside from humans, it is largely rodents and most notably rabbits that have an appendix. Therefore, using rabbits as my example:

In rabbits, the appendix is thought to have a key role in the development of the immune system. Specifically it has been shown experimentally that when neonatal rabbits are given an appendectomy levels of Immunoglobulin A and G (IgA/IgG) fall dramatically. Both these polypeptides are prominent antibodies - IgA plays a key role in mucosal immunity whilst IgGs in humoral immunity. These effects were localised to the small intestine of the rabbits, however were statistically significant. 1

In infants, the rabbit appendix resembles the chicken bursa and sheep ileal Peyer's patch (both performing similar functions as described above).2

This has led to some ongoing research as to whether the human appendix has a similar properties in having function in infants 3:

If the human appendix functions as a primary lymphoid organ, it may occur during the first few months of age when the GC T-cell density is low.